398 VIII. NEWTONIAN POTENTIAL FUNCTION. 



145. Harmonics in Spherical Coordinates. We have trans- 

 formed Laplace's Operator into spherical coordinates in 135, and 

 =0 becomes 



If we put in this V n r n Y n we obtain 

 99) sintf. (+!) T n + A 



as the differential equation satisfied by a surface harmonic Y n (& t qp). 

 This is the form of Laplace's equation originally given by Laplace. 1 ) 

 If Y n is the zonal harmonic P nj which is independent of cp, we have 



100) 



or putting 



cos # = 



This is known as Legendre's Differential Equation. We shall 

 now, without considering more in detail the general surface harmonic, 

 find the general expression for the zonal harmonic. It may be at 

 once shown, by inserting for P(/A) a power- series in /* and deter- 

 mining the coefficients, that for integral values of n the differential 

 equation is satisfied by a polynomial in p. The form of these 

 polynomials we shall find from one of their important properties. 



146. Development of Reciprocal Distance. We know 

 that ; the reciprocal of the distance of the point x, y, z from any 



fixed point P, is a harmonic function of the coordinates x, y, 0, and 

 although it is not a homogeneous function except when the fixed 

 point is the origin, it may always be developed in a series of homo- 

 geneous functions, that is, in a series of spherical harmonics. We 

 shall now use the letter d for the distance from any fixed point, 

 reserving r for the distance from the origin. Let us for convenience 

 take the axis of as passing through the fixed point P, which lies 

 at a distance r' from the origin, and put cos (r0) = /i. Then we have 



102) 1 = [r 2 + r ' 2 - 2rr >]~ ^ = [> 2 + if 



1) Laplace, "Theorie des attractions des sphero'ides et de la figure des 

 planetes." Mem. de I'Acad. de Paris. Annee 1782 (publ. 1785). 



